Consider a Toeplitz matrix T which has both positive and negative eigenvalues. Can we prove that there exist two Toeplitz matrix T1 and T2 such that T1+T2=T and T1 has only one positive Eigenvalues and T2 has only negative Eigenvalues? Any reply would be appreciated.
Regards

The Toeplitz matrix considered is Hermitian matrix. Thank you for looking at my question.
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RantuAug 30 '12 at 23:41

1

I think you need further conditions to make this interesting. You can take $T1 = T + \gamma I$, $T2 = - \gamma I$, where $\gamma$ is a large constant. This way, $T1$ will have only positive eigenvalues, and $T2$ will have negative ones (since you're shifting them by $\gamma$).
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comaAug 30 '12 at 23:59